In learning by imitation, the correspondence problem is that of identifying what action sequence of the imitator is "closest" to that of the demonstrator. However, the notion of "close" has largely remained fuzzy so far, and without any objective measure. In this project, we are developing a metric that can provide a scalar measure of dissimilarity between the actions of agents, with possibly different kinematic structures. By doing so we intend to provide a generic means to evaluate imitation across dissimilar embodiments.

Agents are treated as kinematic chains, and are represented in the form of a tree termed a kinematic tree or just k-tree.

Definition (k-tree):A kinematic tree or k-tree is an encoding of an open kinematic chain such that every link in the kinematic chain is represented by a unique edge in the k-tree.
The configuration of the chain is encoded into the k-tree by introducing the notion of a pose.
Definition (pose):A pose p of a k-tree t is an assignment of the ordered pair <ni,
li> to every edge ei of t. ni is a unit
normal that is the orientation of the link, represented by ei,
in the world coordinate frame. li is the length of the link represented
by ei.
The set of all poses is partitioned into pose classes based on the structure of their respective k-trees.
Definition (pose class):A
pose class P is a set of poses such that for any pair
of poses p, q € P, if tp is the k-tree of p and tq
the k-tree of q then tp and tq are homeomorphic.
We have developed an algorithm that provides a distance measure between any pair of poses belonging to the same pose class. The algorithm proceeds by comparing corresponding segments of the respective k-trees to get partial measures of dissimilarity and summing these values to obtain the distance between the poses. This distance measure is invariant to rotations of the individual poses. It is also scale invariant. The distance also satisfies the following triangular inequality.
distance(p, q) + distance(q, r) ≥ distance(p, r)

These properties together make this distance measure a pseudometric. Thus, pose distance imposes a pseudometric on a pose class.

The pose distance was applied to the poses of three different agents, namely, a human, the Sony AIBO dog robot, and a simulated dolphin-like skeleton. The metric can be applied to the poses of these agents, because their poses belong to the same pose class. Figure 1 shows an example pose for each of these agents.

Figure 1: Poses of the human, the Sony AIBO and the dolphin-like skeleton.

The pairwaise distances of various poses of these agents were computed and a symmetric dissimilarity matrix was created. The Multi-dimensional Scaling (MDS) algorithm was applied on this dissimilarity matrix and an embedding on a 2D Euclidean space was calculated. Figure 2 shows the embedding of all the poses. It can be seen that similar poses are closer to each other than less similar ones.

Figure 2: Embedding of poses on a 2D Euclidean space.

We are currently pursuing the extension of this metric to the comparison of actions.